Asked by Jayden
Compare Earth and Mercury's period of revolution around the sun. How can you explain the differences in length
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Answered by
GPT-5 mini
Mercury takes about 88 Earth days to revolve once around the Sun; Earth takes about 365.25 days. So Mercury’s orbital period is much shorter.
Why? Kepler’s third law (and Newtonian gravity) explain it:
- Kepler’s third law: T^2 ∝ a^3, where T is the orbital period and a is the orbit’s semi‑major axis (average distance from the Sun).
- In Newtonian form: T = 2π sqrt(a^3 / GM☉). Because the Sun’s mass GM☉ is the same for both planets, the period depends only on distance a.
Numeric illustration using astronomical units (1 AU = Earth–Sun distance):
- Mercury’s a ≈ 0.387 AU, so TMercury ≈ 0.387^(3/2) years ≈ 0.241 years ≈ 88 days.
- Earth’s a = 1 AU, TEarth = 1 year ≈ 365.25 days.
Physical intuition: closer to the Sun the gravitational pull is stronger, so a planet must move faster to remain in orbit. Mercury is much closer than Earth, so it travels a smaller orbit and at a much higher orbital speed — both factors make its orbital period much shorter. (Mercury’s orbit is also more eccentric than Earth’s, so its instantaneous speed and distance vary more over an orbit, but the main reason for the shorter period is its smaller semi‑major axis.)
Why? Kepler’s third law (and Newtonian gravity) explain it:
- Kepler’s third law: T^2 ∝ a^3, where T is the orbital period and a is the orbit’s semi‑major axis (average distance from the Sun).
- In Newtonian form: T = 2π sqrt(a^3 / GM☉). Because the Sun’s mass GM☉ is the same for both planets, the period depends only on distance a.
Numeric illustration using astronomical units (1 AU = Earth–Sun distance):
- Mercury’s a ≈ 0.387 AU, so TMercury ≈ 0.387^(3/2) years ≈ 0.241 years ≈ 88 days.
- Earth’s a = 1 AU, TEarth = 1 year ≈ 365.25 days.
Physical intuition: closer to the Sun the gravitational pull is stronger, so a planet must move faster to remain in orbit. Mercury is much closer than Earth, so it travels a smaller orbit and at a much higher orbital speed — both factors make its orbital period much shorter. (Mercury’s orbit is also more eccentric than Earth’s, so its instantaneous speed and distance vary more over an orbit, but the main reason for the shorter period is its smaller semi‑major axis.)
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